How do we export a single object, multiple objects, or selected layers?

@LemonPeeler ,

It's not easy to understand your question because you are using two words which don't describe what you mean when dealing with Elements.

Object:

In other softwares like Illustrator or Indesign, objects are clearly defined as graphical components of an image. There is no precise meaning of objects in Elements.  It's "anything" in the image, in its components, its properties.

Export:

If you are curious about what that means for Photoshop, have a look at the hundreds of discussions following a recent change in the 'save', 'save as' or 'export' commands. We don't suffer the same issue with Elements, but really "Export" is only significant for the organizer, not for the editor.

"Layers" are clearly defined in Elements, and Greg has given a good answer about moving them between opened files. If you mean saving selected layers into new files, that's also possible. Thanks for giving more precisions.

Export to where?  Another open image?  If so (and if you do not have floating windows enabled), select Tile from the Window>Images> menu.  This will open up your two (?) images side by side.  Select your multiple layers and click and drag the layers to the other image.